To the extent that you can hold a heart in your hand, it differs from the mind and the soul (assuming that mind does not equal brain). So you wish to talk about "heart" in the abstract. The soul seems to be an intangible concept by definition. The mind means different things to philosophers, psychologists, and biologists.
If it's not too sensitive an issue, think about the "strategy" of "winning the hearts and minds of the Iraqui people." These are definitely abstract nouns. We're speaking of the heart and the mind of an individual person - one heart and one mind. I think that's a countable noun. If you "win over" three people, that's three hearts and three minds. It's still abstract.
The heart of the American people is singular, and very abstract, and surely has a different meaning than "winning the heart of my girlfriend." Is it uncountable? Personally, I don't think so. "One" is a number. It's possible to "win the heart of the American people."
Do you have a reference saying that abstract nouns are uncountable? I don't think "intangible" = "uncountable." Ideas are intangible. I came up with four new ideas today. I think that "abstract" and "concrete" are both members of the common noun classification. Asphalt and concrete are examples of uncountable nouns.
Regards, - A.